Theorem inimass 6015

Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
inimass	⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶))

Proof of Theorem inimass

Step	Hyp	Ref	Expression
1		rnin 6008	. 2 ⊢ ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) ⊆ (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶))
2		df-ima 5571	. . 3 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐵) ↾ 𝐶)
3		resindir 5873	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
4	3	rneqi 5810	. . 3 ⊢ ran ((𝐴 ∩ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
5	2, 4	eqtri 2847	. 2 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
6		df-ima 5571	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
7		df-ima 5571	. . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
8	6, 7	ineq12i 4190	. 2 ⊢ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶))
9	1, 5, 8	3sstr4i 4013	1 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶))

Syntax hints:   ∩ cin 3938   ⊆ wss 3939  ran crn 5559   ↾ cres 5560   “ cima 5561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571

This theorem is referenced by:  restutopopn  22850